The classical action \(S^{(0)}\)  and the Chern-Simons term in \(S^{(1)}\)  give the description of a version of the Son's theory [1]}
of the Dirac composite fermion [2]}, [3]}, [4]} that assumes the Pauli-Villars type of regularization [5]}, [6]} because
of the presence of the CS term for field \(a_\mu \) . Note that the CS term has the correct coefficient, \(\tfrac{1}{8\pi }\) . Also, in the linear NC correction \(S^{(1)}\)  the Dirac momentum density couples to the external electric field as expected from the Galilean invariance [7]}, [8]}. On the other hand, the presence of a coupling to the internal electric field \(\hat{a}_\mu \)  is quite natural and expected given the influence of other particles on a selected one. Also, we find that the presence of internal and external (i.e., departure
from the uniform) magnetic field induces a change in the coefficient of the kinetic terms \(\hat{c}^\dagger \partial _\tau \hat{c}\)  and \(\hat{d}^\dagger \partial _\tau \hat{d}\) . Thus, we can conclude that the NC formulation, up to the first order in \(\theta \) , recovers known results but also systematically adds terms that we may expect on physical grounds.
